Let us still consider the problem (REF ) with \(\varepsilon =0.01\) .
We adopt the uniform spatial mesh with \(h=1/512\)  and give the initial value by random numbers between \(-0.8\)  and \(0.8\)  on each mesh point. We then set the time step size \({\tau }=0.01\) , and compute the numerical solutions by using the sESAV schemes, the classic SAV schemes (SAV1 and SAV2), and the ESAV schemes (ESAV1 and ESAV2).
Note that we set \(\delta =C_0+0.01\) 
for the classic SAV schemes (REF ) and (REF ).
For all comparison experiments, we will consider two settings for the stabilizing parameter: \(\kappa =\Vert f^{\prime }\Vert _{C[-\beta ,\beta ]}\)  and \(\kappa =\frac{1}{2}\Vert f^{\prime }\Vert _{C[-\beta ,\beta ]}\) ,
where the former one satisfies the requirement for the MBP preservation for the sESAV schemes
and the latter one was adopted in [1]} for the classic SAV schemes.
In addition, we take the numerical results obtained by the IFRK4 scheme [2]}
with the small time step size \(10^{-4}\)  as the benchmark solution.
